Question

What is happening to the electron relative to the nucleus when ?E=0? ?E>0? ?E<0?

Answer 1

We set the zero potential energy level at r --> infinity. And that makes anything less than that distance from the core of an atom, negative PE. So the inner most, n = 1, shell is the most negative stored energy of them all. And of course, that means the PE for the n = 2 shell is less negative (more positive) than the n = 1 PE.

In classical electrostatics, PE = - qKQ/r where Q is the proton charge(s) and q is the electron. As you can see, as r --> inf, PE --> 0 from the negative side. The quantum relationship, with the Rydberg number, Planck's Constant, etc., follows the same pattern. So, indeed, the potential energies of electrons increase as the shell numbers increase.

So there's the inner most electron sitting at the very bottom of a negative potential energy well. To escape that well and break the bond between it and the nucleus, the electron must have kinetic energy KE > PE for a complete breakaway. And as that's a very very deep well in close and personal to the nucleus, that's unlikely to happen. Conversely, electrons on the outer shells have smaller negative PE; so they don't need much kinetic energy to break totally free from the atom. And that's why, chemistry major, the valence electrons tend to be the outer ones. They're easier to share out there.

But, this is a big BUT, the electron could hop up to the bottom of the next lower PE well. That is, they don't need to break away; they can simply move up a shell or two. And for that, the amount of kinetic energy would be ke < KE. That is, the kinetic energy to hop up to the next level or more would be less than the break away energy requirement.

So when a photon comes calling on that n = 1 electron, it pays it way by giving up its photon energy e = hF = ke which causes the electron to hop up to n = 2 (if there's an open orbital there). So at n = 2, the electron does indeed have higher kinetic energy than when it was at n = 1. It has to have; so it can jump to a lesser negative (more positive) potential energy well. As an image, think of the electron in that very deep well jumping up to a lower, less deep ledge inside that well.

So we can say that ke = E1 < E2 = ke + hF = KE are the relative kinetic energies at the En shell. And that difference E2 - E1 = e = hF is the added kinetic energy from the coupled photon with frequency F. QED.

And there you are. It all starts with that negative potential energy well.